3.6 Using Exponentials to Approximate General Times 87
3.6.2 Erlang Inter-Arrival Times
If the inter-arrival process is an Erlang distribution then the state-space scheme is
slightly different from that used for Erlang service. The same concept of breaking
the service process into phases is used for the arrival process; however, the state
space will be slightly different. We illustrate the expanded state space process ap-
plied to arrivals by assuming an Erlang-2 inter-arrival time process. The arrivals will
be processed one-at-a-time at a single workstation with exponentially distributed
service times with a limit of three jobs in the system, in other words, we consider an
E
2
/M/1/3system.
Conceptually, an arriving job is always in one of two phases, and each phase has
a mean rate of 2
λ
or a mean sojourn time of 1/(2
λ
). As long as a job is in one of
the arrival phases, it is not yet considered part of the system. The arriving job begins
in phase 1. After an exponentially distributed length of time, the job transitions to
phase 2. After another exponential length of time, two events occur simultaneously:
the job leaves phase 2 and enters the system and another jobs enters phase 1. (Note
that for a model of phased arrivals, one of the arrival phases is always occupied and
the other phases are empty.)
The slight difference in the state space for the Erlang inter-arrival time model
versus the Erlang service time model occurs due to the situation that the arrival
process has two phases regardless of the number of jobs in the system. So when the
system is empty, there are still two phases that the arriving job must complete before
it becomes an active job attempting to enter the system. The state-space notation
used is (i,n) where as before i is the phase and n is the number of jobs in the
system. Note that the order has been reversed from the Erlang service model to help
keep in mind that the phases are for the arrival process. The states needed to model
the E
2
/M/1/3systemare:{(1,0), (2,0), (1,1), (2,1), (1,2), (2,2), (1,3), (2,3)}.The
diagram of this model is given in Fig. 3.3. Note also that there is a different situation
for blocked jobs for this model. A job is not blocked until it arrives to a full system
which occurs from state (2,3) with rate 2
λ
. Then the arrival process starts over in
state (1,3) rather than staying at state (2,3). That is, the arriving job is rejected and
the arrival process starts over at state (1,3) for the next job creation. Thus, there is
an arc between (2,3) and (1,3) with rate 2
λ
in Fig. 3.3 to represent this transition.
Instead of using cuts to derive the equations of state, we use the single-node iso-
lation method for generating the equations that define the steady-state probabilities.
The following system of equations (all eight equations are given but only seven are
used since the norming equation is also required) are generated for the states in the
order that they appear in the above state list.
2
λ
p
10
=
μ
p
11
2
λ
p
20
= 2
λ
p
10
+
μ
p
21
(2
λ
+
μ
)p
11
= 2
λ
p
20
+
μ
p
12
(2
λ
+
μ
)p
21
= 2
λ
p
11
+
μ
p
22